const In : set set prop term iIn = In infix iIn 2000 2000 const PNo_upc : (set (set prop) prop) set (set prop) prop const PNoLt : set (set prop) set (set prop) prop term PNo_rel_strict_lowerbd = \P:set (set prop) prop.\x:set.\p:set prop.!y:set.y iIn x -> !q:set prop.PNo_upc P y q -> PNoLt x p y q term Subq = \x:set.\y:set.!z:set.z iIn x -> z iIn y term TransSet = \x:set.!y:set.y iIn x -> Subq y x term ordinal = \x:set.TransSet x & !y:set.y iIn x -> TransSet y lemma !P:set (set prop) prop.!x:set.!p:set prop.!y:set.!z:set.!q:set prop.y iIn x -> TransSet x -> TransSet y -> (!w:set.w iIn x -> !p2:set prop.PNo_upc P w p2 -> PNoLt x p w p2) -> z iIn y -> PNo_upc P z q -> z iIn x -> PNoLt y p z q var P:set (set prop) prop var x:set var p:set prop var y:set hyp y iIn x hyp TransSet x hyp ordinal y claim TransSet y -> (!z:set.z iIn x -> !q:set prop.PNo_upc P z q -> PNoLt x p z q) -> !z:set.z iIn y -> !q:set prop.PNo_upc P z q -> PNoLt y p z q